# Yang-Baxter $\sigma$-models, conformal twists, and noncommutative Yang-Mills theory

## TL;DR

This paper explores how Yang-Baxter sigma-model deformations of AdS5×S5 can be understood as noncommutative geometries via conformal twists, linking integrability of the sigma-model to the properties of the dual noncommutative gauge theory.

## Contribution

It recasts Yang-Baxter deformations as conformal twists affecting the noncommutative parameter, establishing a connection between integrability conditions and divergence-free noncommutativity.

## Key findings

- Deformations are encoded in a divergence-free noncommutative parameter Θ.
- Unimodularity of r-matrices ensures supergravity solutions and integrability.
- Dual noncommutative gauge theories are planar integrable.

## Abstract

The Yang-Baxter $\sigma$-model is a systematic way to generate integrable deformations of AdS$_5\times$S$^5$. We recast the deformations as seen by open strings, where the metric is undeformed AdS$_5\times$S$^5$ with constant string coupling, and all information about the deformation is encoded in the noncommutative (NC) parameter $\Theta$. We identify the deformations of AdS$_5$ as twists of the conformal algebra, thus explaining the noncommutativity. We show that the unimodularity conditon on $r$-matrices for supergravity solutions translates into $\Theta$ being divergence-free. Integrability of the $\sigma$-model for unimodular $r$-matrices implies the existence and planar integrability of the dual NC gauge theory.